k (i.e., the game-hypergraph is k-uniform), then the Erdős-Selfridge theorem implies that Breaker wins whenever , i.e., the number of winning-sets is less than . The number is tight: there are -uniform hypergraphs where the number of winning sets is exactly , and where Maker has a winning strategy. For example, consider a perfect binary tree of height . It has leaves. Define V as the set of tree nodes, and H as the family of all paths from the root to a leaf. Maker starts by picking the root. Then, if Breaker picks an element in the left subtree, Maker picks the root of the right subtree, and vice versa. By continuing this way, Maker can always pick a full path, i.e., a winning-set. Disjoint and almost-disjoint hypergraphs If all winning-sets are pairwise-disjoint and their size is at least 2, then Breaker can win using a pairing strategy. Suppose now that the winning-sets are almost disjoint, i.e., any two winning-sets have at most one element in common. If all winning-sets are of size , and the number of winning sets is less than (for some fixed constant c), then Breaker has a winning strategy. So this situation is easier for Breaker than the general case, but harder than the case of disjoint winning-sets. A winning condition for Maker Define the degree of a set of elements as the number of different winning-sets that contain this set. Define the pair-degree of a set-family, denoted , as the maximum degree of a pair of elements (maximum over all pairs). If all winning-sets are of size , and the number of winning-sets is more than , then Maker has a winning strategy. The strategy uses the same potential function used by Erdos and Selfridge: the potential of a winning-set with unoccupied elements (and no element occupied by Breaker) is . The value of an element is the total potential-decrease if Breaker takes that element, which is the same as the total potential-increase if Maker takes it. Maker's strategy is simply to take the highest-value element. Whenever Maker takes an element, the potential of every winning-set that contains it increases by ; whenever Breaker takes an element, the potential of every set that contains it and does not contain Maker's element decreases by . Therefore, if we consider only the winning-sets that were touched once, the potential-sum weakly increases. The potential-sum can decrease